Beginning his fourth season with the Browns, Couch played the entire first half and looked sharp, tossing a 45-yard touchdown pass to Jamel White just 2:05 into the contest.
Moe Williams, Minnesota's leading rusher with 19 yards on four carries, tied the game on a four-yard run with 4:58 left in the first quarter. But Phil Dawson kicked field goals of 46 and 45 yards in the second quarter to give the Browns a 13-7 halftime lead.
Cleveland took a 20-7 lead on Kalvin Pearson's 62-yard interception return 4:14 into the second half.
Kelly Holcomb, who completed seven of 10 for 100 yards, threw a five-yard touchdown strike to Mark Campbell to make it 27-7 with 7:12 left in the third period.
Autry Denson paced Cleveland on the ground with 41 yards on eight carries. William Green, the Browns' first-round pick, rushed for just 11 yards on six carries in his NFL debut.
Minnesota played without offensive tackle Bryant McKinnie, its first-round pick who remains unsigned.
Minnesota got its final touchdown when Matt Birk recovered a fumble in the end zone with 2:37 left in the contest.
2014: The Year in Music [PHOTOS]